In this paper, we develop a new notion of prediction validity that goes beyond the coverage probability of prediction regions, one that aims to assess the performance of a predictive distribution on these other relevant tasks. If a predictive distribution is to be used to evaluate the probability of certain assertions about the future observation, then the magnitude of this probability will be used to draw inference about whether the assertion will be true. So, roughly, to avoid making systematically erroneous predictions, it would be desirable to control the rate at which the predictive distribution assigns small probability to assertions that happen to be true. Our definition of strong prediction validity in Section REF —compared to prediction region coverage probabilities, which we refer to as weak prediction validity in Section REF —makes this notion of avoiding systematically erroneous predictions precise. There are a number of important imprecise probability-related consequences of strong validity. First, strong validity rules out the possibility of a sure loss, forging an interesting connection between the classical behavioral interpretation of imprecise probabilities and the statistical properties of a procedure based on them. Second, we argue that strong prediction validity cannot be achieved by any additive/precise predictive probability distribution that does not have access to the unknown true distribution. Therefore, if one agrees that our proposed notion of strong prediction validity is important, then he must seek out an appropriate imprecise probability to serve his prediction uncertainty quantification needs. Compare this to the conclusions drawn from the false confidence theorem [1]}, [2]} in the context of statistical inference.
